Finite Group Databasebeta

← #27145002

Subgroups — #27145002

Order 12416.

560 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 560 conjugacy classes of subgroups — too many to draw. Only its 143 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15690772C211yesmaximal, characteristic
2C2×#15690768C211yesmaximal, characteristic, complemented
3#20260670C211yesmaximal, characteristic, complemented
4#20260665C211yesmaximal, characteristic, complemented
5#20260664C212yesmaximal, complemented
6#20260662C211yesmaximal, characteristic, complemented
7#20260653C212yesmaximal
8#20260586C212yesmaximal, complemented
9#20260582C212yesmaximal, complemented
10#20260582C211yesmaximal, characteristic, complemented
11#400×C97C211yesmaximal, characteristic, complemented
12C22×D388C2211yescharacteristic
13C22×C4×D97C2211yescharacteristic
14C22×C4×D97C2211yescharacteristic
15#15690772—24no—
16#15690768—24no—
17C2×C4⋊Dic97C2211yescharacteristic
18C2×#13095490C2212yes—
19C2×#13095490C2212yes—
20C2×#13095488C2212yes—
21C2×#13095488C2212yes—
22C2×#13095488C2212yes—
23C2×#13095488C2212yes—
24C2×#13095488C2211yescharacteristic
25C2×#13095488C2211yescharacteristic
26C2×C4×Dic97C2212yes—
27C2×C4×Dic97C2211yescharacteristic
28C2×C4×Dic97C2212yes—
29#15690717C2212yes—
30#15690717C2212yes—
31#15690717C2211yescharacteristic
32C2×C4⋊C4×C97C2211yescharacteristic
33C2×C4⋊C4×C97C2211yescharacteristic
34#95×C97C2212yes—
35#95×C97C2211yescharacteristic
36#95×C97C2212yes—
37C23×D97C2311yescharacteristic
38C23×D97C2311yescharacteristic
39C2×D388—22no—
40C2×D388—22no—
41C2×D388—22no—
42C22×Dic97C2311yescharacteristic
43C22×Dic97C2312yes—
44C22×Dic97C2312yes—
45C22×Dic97C2311yescharacteristic
46C2×C4×D97—24no—
47C2×C4×D97—22no—
48C2×C4×D97—24no—
49C2×C4×D97D414yes—
50C4⋊Dic97—22no—
51#13095490—24no—
52#13095490—24no—
53#13095488—24no—
54#13095488—22no—
55#13095488—24no—
56#13095488—22no—
57#13095488—24no—
58#13095488—24no—
59C4×Dic97—24no—
60C4×Dic97—22no—
61C4×Dic97—24no—
62C22×C4×C97C2312yes—
63C22×C4×C97C2312yes—
64C22×C4×C97C2311yescharacteristic
65C22×C4×C97C2311yescharacteristic
66C22×C4×C97C2311yescharacteristic
67C4⋊C4×C97—22no—
68C4⋊C4×C97—22no—
69C23×C97C2411yescharacteristic
70C22×D97—21no—
71C22×D97—22no—
72C22×D97—22no—
73C22×D97—22no—
74C22×D97—22no—
75C22×D97—21no—
76C22×D97—21no—
77C22×D97—21no—
78C22×D97C2×D412yes—
79C22×D97#4112yes—
80C2×C4×C97—21no—
81C2×C4×C97—22no—
82C2×C4×C97—21no—
83C2×C4×C97—22no—
84C2×C4×C97—21no—
85C2×C4×C97—22no—
86C2×C4×C97—22no—
87C2×C4×C97—22no—
88C2×C4×C97—22no—
89C2×C4×C97—22no—
90C2×C4×C97—22no—
91C2×C4×C97C2×D412yes—
92C2×C4×C97#4112yes—
93C2×Dic97—22no—
94C2×Dic97—22no—
95C2×Dic97—21no—
96C2×Dic97—22no—
97C2×Dic97—22no—
98C2×Dic97—22no—
99C2×Dic97—22no—
100C2×Dic97C2×D412yes—
101C2×Dic97#4114yes—
102C2×Dic97#4114yes—
103D388—28no—
104C4×D97—28no—
105C4×D97—44no—
106C22×C97C22×D411yescharacteristic
107C22×C97C2×#4112yes—
108C22×C97C2×#4111yescharacteristic
109C22×C97C2×#4112yes—
110C22×C97C2×#4111yescharacteristic
111C2×D97—22no—
112C2×D97—22no—
113C2×D97—22no—
114C2×D97—24no—
115C2×D97—22no—
116C2×D97—41no—
117C2×D97—42no—
118C2×D97—42no—
119C2×D97—42no—
120C2×D97—41no—
121C4×C97—22no—
122C4×C97—22no—
123C4×C97—41no—
124C4×C97—42no—
125C4×C97—42no—
126Dic97—24no—
127Dic97—42no—
128Dic97—41no—
129Dic97#12314yes—
130C2×C97C2×#12311yescharacteristic
131C2×C97#52511yescharacteristic
132C2×C97#52912yes—
133C2×C97#52912yes—
134C2×C97#53211yescharacteristic
135D97—42no—
136D97—42no—
137#2395—971nomaximal, Sylow
138C97#239511yescharacteristic, complemented, Sylow
139C2×C4×D4—971no—
140C2×#117—971no—
141#400D9711yescharacteristic, complemented
142#399—971no—
143#398—971no—
144#397—971no—
145#397—972no—
146#394—972no—
147C4×C4⋊C4—972no—
148C4×C22⋊C4—972no—
149C4×C22⋊C4—971no—
150C22×D4—971no—
151C23×C4—971no—
152C23×C4—971no—
153C4×D4—1944no—
154#117—1944no—
155C2×C4⋊C4—971no—
156C2×C4⋊C4—972no—
157C2×C4⋊C4—972no—
158C2×C4⋊C4C2×D9711yescharacteristic
159C2×C4⋊C4C2×D9711yescharacteristic
160C2×C22⋊C4—972no—
161C2×C22⋊C4—971no—
162C2×C22⋊C4—971no—
163C2×C22⋊C4—972no—
164C2×C22⋊C4—972no—
165C2×C22⋊C4—972no—
166C2×C42—972no—
167C2×C42—971no—
168C2×C42—972no—
169#95—972no—
170#95—971no—
171#95—972no—
172#95C2×D9712yes—
173#95C2×D9711yescharacteristic
174#95C2×D9712yes—
175C24—971no—
176C24—971no—
177C2×D4—1942no—
178C2×D4—1942no—
179C2×D4—1942no—
180C22×C4—971no—
181C22×C4—972no—
182C22×C4—974no—
183C22×C4—971no—
184C22×C4—972no—
185C22×C4—1942no—
186C22×C4—1944no—
187C22×C4—1944no—
188C22×C4C22×D9711yescharacteristic
189C22×C4C22×D9712yes—
190C22×C4C22×D9712yes—
191C22×C4C22×D9711yescharacteristic
192C22×C4C22×D9711yescharacteristic
193C4⋊C4—22no—
194C4⋊C4—22no—
195C4⋊C4—1944no—
196C4⋊C4—1942no—
197C4⋊C4—1944no—
198C22⋊C4—1944no—
199C22⋊C4—1944no—
200C22⋊C4—1944no—
201C22⋊C4—1944no—
202C22⋊C4—1942no—
203C22⋊C4—1942no—
204C42—1944no—
205C42—1944no—
206C42—1942no—
207C23—972no—
208C23—972no—
209C23—1941no—
210C23—1942no—
211C23—1942no—
212C23—1942no—
213C23—1942no—
214C23—1941no—
215C23—1941no—
216C23—1941no—
217C23C23×D9711yescharacteristic
218D4—1948no—
219C2×C4—21no—
220C2×C4—22no—
221C2×C4—21no—
222C2×C4—22no—
223C2×C4—21no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—972no—
231C2×C4—974no—
232C2×C4—974no—
233C2×C4—3884no—
234C2×C4—1948no—
235C2×C4—1941no—
236C2×C4—1942no—
237C2×C4—1942no—
238C2×C4—1942no—
239C2×C4—1942no—
240C2×C4—1942no—
241C2×C4—1942no—
242C2×C4D4×D9712yes—
243C2×C4#1309551212yes—
244C22—3881no—
245C22—3882no—
246C22—3882no—
247C22—3882no—
248C22—3881no—
249C22—1942no—
250C22—1942no—
251C22—1942no—
252C22—1944no—
253C22—1942no—
254C22C2×D4×D9711yescharacteristic
255C22#41×D9712yes—
256C22#41×D9711yescharacteristic
257C22#41×D9712yes—
258C22C2×#1309551211yescharacteristic
259C4—22no—
260C4—22no—
261C4—41no—
262C4—42no—
263C4—42no—
264C4—974no—
265C4—3882no—
266C4—3881no—
267C4—1944no—
268C2—3882no—
269C2—3882no—
270C2#123×D9711yescharacteristic
271C2#2026124811yescharacteristic
272C2#2026126812yes—
273C2#2026127012yes—
274C2#2026128411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D97
commutator, socle, cosocleC23×C97C24
Fitting#400×C97C2
prime radicalC22×D388C22
hypercenter#400D97
hypocenterC97#2395
LCSG ▷ C23×C97 ▷ C97
UCSC1 ◁ C23 ◁ #400

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×#15690768C2semidirect
3. #20260670C2semidirect
4. #20260665C2semidirect
5. #20260664C2semidirect
6. #20260662C2semidirect
8. #20260586C2semidirect
9. #20260582C2semidirect
10. #20260582C2semidirect
11. #400×C97C2semidirect
138. C97#2395semidirect
141. #400D97semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#156907722
C2×#156907682
#202606702
#202606652
#202606642
#202606622
#202606532
#202605862
#202605822
#400×C972
#239597